Google service to replace: Inbox by Gmail
Thunderbird is a free, open source, cross-platform e-mail, news and instant messaging client. The project strategy is modelled after Mozilla Firefox. It supports POP (storing mail locally on your hard drive so that it can be accessed without an internet connection), and IMAP. It features excellent mail filter capabilities and management. Thunderbird has strong support for using multiple accounts and identities, including automated signature features. Thunderbird can also function as an RSS/Atom feed reader, and has a large selection of add-ons that extend it with calendar support (Lightning Calendar), PGP support (Enigmail), integration with online services and much more.
Also worth mentioning: Spark, Astro, Canary Mail, K-9 Mail, Claws Mail & Mailspring.
PS : One of the reasons I didn't use Thunderbird before was because of its UI which is pretty old and ugly. But thanks to that slick Thunderbird theme, you can now use it with a clean and nice interface.